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Theorem dcomex 10484
Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)
Assertion
Ref Expression
dcomex ω ∈ V

Proof of Theorem dcomex
Dummy variables 𝑡 𝑠 𝑥 𝑓 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 8524 . . . . . . 7 1o ≠ ∅
2 df-br 5148 . . . . . . . 8 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩})
3 elsni 4647 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩)
4 fvex 6919 . . . . . . . . . 10 (𝑓𝑛) ∈ V
5 fvex 6919 . . . . . . . . . 10 (𝑓‘suc 𝑛) ∈ V
64, 5opth1 5485 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩ → (𝑓𝑛) = 1o)
73, 6syl 17 . . . . . . . 8 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → (𝑓𝑛) = 1o)
82, 7sylbi 217 . . . . . . 7 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → (𝑓𝑛) = 1o)
9 tz6.12i 6934 . . . . . . 7 (1o ≠ ∅ → ((𝑓𝑛) = 1o𝑛𝑓1o))
101, 8, 9mpsyl 68 . . . . . 6 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1o)
11 vex 3481 . . . . . . 7 𝑛 ∈ V
12 1oex 8514 . . . . . . 7 1o ∈ V
1311, 12breldm 5921 . . . . . 6 (𝑛𝑓1o𝑛 ∈ dom 𝑓)
1410, 13syl 17 . . . . 5 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
1514ralimi 3080 . . . 4 (∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
16 dfss3 3983 . . . 4 (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
1715, 16sylibr 234 . . 3 (∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
18 vex 3481 . . . . 5 𝑓 ∈ V
1918dmex 7931 . . . 4 dom 𝑓 ∈ V
2019ssex 5326 . . 3 (ω ⊆ dom 𝑓 → ω ∈ V)
2117, 20syl 17 . 2 (∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ∈ V)
22 snex 5441 . . 3 {⟨1o, 1o⟩} ∈ V
2312, 12fvsn 7200 . . . . . . . 8 ({⟨1o, 1o⟩}‘1o) = 1o
2412, 12funsn 6620 . . . . . . . . 9 Fun {⟨1o, 1o⟩}
2512snid 4666 . . . . . . . . . 10 1o ∈ {1o}
2612dmsnop 6237 . . . . . . . . . 10 dom {⟨1o, 1o⟩} = {1o}
2725, 26eleqtrri 2837 . . . . . . . . 9 1o ∈ dom {⟨1o, 1o⟩}
28 funbrfvb 6961 . . . . . . . . 9 ((Fun {⟨1o, 1o⟩} ∧ 1o ∈ dom {⟨1o, 1o⟩}) → (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o))
2924, 27, 28mp2an 692 . . . . . . . 8 (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o)
3023, 29mpbi 230 . . . . . . 7 1o{⟨1o, 1o⟩}1o
31 breq12 5152 . . . . . . . 8 ((𝑠 = 1o𝑡 = 1o) → (𝑠{⟨1o, 1o⟩}𝑡 ↔ 1o{⟨1o, 1o⟩}1o))
3212, 12, 31spc2ev 3606 . . . . . . 7 (1o{⟨1o, 1o⟩}1o → ∃𝑠𝑡 𝑠{⟨1o, 1o⟩}𝑡)
3330, 32ax-mp 5 . . . . . 6 𝑠𝑡 𝑠{⟨1o, 1o⟩}𝑡
34 breq 5149 . . . . . . 7 (𝑥 = {⟨1o, 1o⟩} → (𝑠𝑥𝑡𝑠{⟨1o, 1o⟩}𝑡))
35342exbidv 1921 . . . . . 6 (𝑥 = {⟨1o, 1o⟩} → (∃𝑠𝑡 𝑠𝑥𝑡 ↔ ∃𝑠𝑡 𝑠{⟨1o, 1o⟩}𝑡))
3633, 35mpbiri 258 . . . . 5 (𝑥 = {⟨1o, 1o⟩} → ∃𝑠𝑡 𝑠𝑥𝑡)
37 ssid 4017 . . . . . . 7 {1o} ⊆ {1o}
3812rnsnop 6245 . . . . . . 7 ran {⟨1o, 1o⟩} = {1o}
3937, 38, 263sstr4i 4038 . . . . . 6 ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}
40 rneq 5949 . . . . . . 7 (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 = ran {⟨1o, 1o⟩})
41 dmeq 5916 . . . . . . 7 (𝑥 = {⟨1o, 1o⟩} → dom 𝑥 = dom {⟨1o, 1o⟩})
4240, 41sseq12d 4028 . . . . . 6 (𝑥 = {⟨1o, 1o⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}))
4339, 42mpbiri 258 . . . . 5 (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 ⊆ dom 𝑥)
44 pm5.5 361 . . . . 5 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
4536, 43, 44syl2anc 584 . . . 4 (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
46 breq 5149 . . . . . 6 (𝑥 = {⟨1o, 1o⟩} → ((𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
4746ralbidv 3175 . . . . 5 (𝑥 = {⟨1o, 1o⟩} → (∀𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
4847exbidv 1918 . . . 4 (𝑥 = {⟨1o, 1o⟩} → (∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
4945, 48bitrd 279 . . 3 (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
50 ax-dc 10483 . . 3 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
5122, 49, 50vtocl 3557 . 2 𝑓𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)
5221, 51exlimiiv 1928 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  Vcvv 3477  wss 3962  c0 4338  {csn 4630  cop 4636   class class class wbr 5147  dom cdm 5688  ran crn 5689  suc csuc 6387  Fun wfun 6556  cfv 6562  ωcom 7886  1oc1o 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753  ax-dc 10483
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570  df-1o 8504
This theorem is referenced by:  axdc2lem  10485  axdc3lem  10487  axdc4lem  10492  axcclem  10494  precsexlem10  28254  seqsex  28305  noseqex  28309
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